Time to failure prediction in rubber components subjected to thermal ageing: A combined approach based upon the intrinsic defect concept and the fracture mechanics

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Mouna BEN HASSINE, M NAÏT-ABDELAZIZ, F ZAÏRI, Xavier COLIN, C TOURCHER, Gregory

MARQUE - Time to failure prediction in rubber components subjected to thermal ageing: A

combined approach based upon the intrinsic defect concept and the fracture mechanics

-Mechanics of Materials - Vol. 79, p.15-24 - 2014

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Time to failure prediction in rubber components subjected

to thermal ageing: A combined approach based upon

the intrinsic defect concept and the fracture mechanics

M. Ben Hassine

a,b

_{, M. Naït-Abdelaziz}

c,⇑

_{, F. Zaïri}

c

_{, X. Colin}

a

_{, C. Tourcher}

b

_{, G. Marque}

b

a

Laboratoire des Procédés et Ingénierie en Mécanique et Matériaux (PIMM), UMR CNRS 8006, Arts et Métiers ParisTech, 151 boulevard de l’Hôpital, F-75013 Paris, France

b

EDF R&D, avenue des Renardières, F-77818 Moret-sur-Loing, France

c

Laboratoire de Mécanique de Lille (LML), UMR CNRS 8107, Université Lille 1 Sciences et Technologies, avenue Paul Langevin, F-59650 Villeneuve d’Ascq, France

Keywords: Rubber Thermal ageing Failure prediction Intrinsic defect Fracture mechanics

a b s t r a c t

In this contribution, we attempt to derive a tool allowing the prediction of the stretch ratio at failure in rubber components subjected to thermal ageing. To achieve this goal, the main idea is to combine the fracture mechanics approach and the intrinsic defect concept. Using an accelerated ageing procedure for an Ethylene–Propylene–Diene Monomer (EPDM), it is first shown that the average molar mass of the elastically active chains (i.e. between cross-links) can be used as the main indicator of the macromolecular network degradation. By introducing the time–temperature equivalence principle, a shift factor obeying to an Arrhe-nius law is derived, and master curves are built as well for the average molar mass as for the ultimate mechanical properties. Fracture mechanics tests are also achieved and the square root dependence of the fracture energy with the average molar mass is pointed out. Moreover, it is shown that the mechanical response could be approximated by the phantom network theory, which allows to relate the strain energy density function to the average molar mass. Assuming that the fracture of a smooth specimen is the conse-quence of a virtual intrinsic defect whose the size can be easily estimated, the stretch ratio at break can be therefore computed for any thermal ageing condition. The estimated values are found in a very nice agreement with EPDM experimental data, making this approach a useful tool when designing rubber components for moderate to high temperature environments.

1. Introduction

The use of rubber components is very widespread in many industrial domains as, for example, in ground or air transportation, medical equipment, electrical insulation, mechanical damping, etc. They can consequently be subjected to complex mechanical loadings in various

environmental conditions. Because of the dependence of the mechanical properties either on the kind of loading or on the operating environment (light, humidity, tempera-ture, oxygen, etc.), an optimal design of these components must account for the alteration of the mechanical properties due to ageing. Indeed, chemical and physical ageings are known to strongly modify the mechanical responses of such materials, but also the ultimate properties. Especially, in the presence of oxygen, the modification of the chemical structure of the elastomeric material is essentially attributed to the macromolecular chain scissions and ⇑Corresponding author.

crosslinking in addition to the break and reformation of crosslink nodes (Rajeev et al., 2003; Rivaton et al., 2005;

Colin et al., 2007a; Tomer et al., 2007). These mechanisms

induce the alteration of the mechanical properties

(Clavreul, 1997; Assink et al., 2002; Colin et al., 2007a).

Chemical ageing of polymers is generally investigated by using analytical techniques allowing the measurement of their chemical, physical and mechanical properties

(Colin et al., 2007a,b,c). Since ageing is a long time process,

accelerated thermal ageing tests are often used to shorten their exposure duration and predict their operating life time. In fact, from these tests, results for lower tempera-tures are generally obtained using the time–temperature equivalence principle (Ferry, 1970; Treloar, 1971; Ha-Anh and Vu-Khanh, 2005; Gillen et al., 2006; Woo and Park,

2011).

The purpose of this contribution is the prediction of stretch ratio at failure in rubbers subjected to thermal age-ing conditions. To achieve this goal, the main idea is to combine the fracture mechanics approach and the intrinsic defect concept. This combined approach1_{was successfully} introduced to study the biaxial fracture of rubbers (

Naït-Abdelaziz et al., 2012) and is extended here in order to

pro-pose a failure criterion accounting for degradation due to thermal ageing.

This paper is organised as follows. In Section2, a gen-eral background on the main tools involved in this paper is presented. It mainly includes fracture mechanics of rubbers, large strain elastic constitutive models and time–temperature equivalence. Material and experimen-tal procedure are detailed in Section 3. In Section 4, experimental results are reported. The predictive approach is presented in Section5: estimations are com-pared to experimental data. Concluding remarks close the paper in Section6.

2. A brief literature background 2.1. Fracture mechanics of rubbers

Since the pioneering work of Griffith (1921), fracture mechanics can generally be tackled via an energy balance which defines a parameter called the strain energy release rate G defined such as:

G ¼ @U

@A ð1Þ

where U is the potential energy and A is the crack area. At crack initiation, the strain energy release rate G takes a critical value generally called the fracture energy and noted Gcwhich is considered as an intrinsic material

prop-erty. Based upon this definition,Rivlin and Thomas (1953)

introduced the tearing energy Tc, equivalent to Gc, in the

case of elastomers. From a simple analysis of the modifica-tion in the stored energy in a specimen containing an edge crack of length a, they expressed Tcin the following form:

Tc¼ Gc¼ 2kðkcÞWca ð2Þ

where Wc is the critical strain energy density and k is a

factor depending on the stretch ratio k = l/l0. The suffix c

denotes the critical value corresponding to crack initiation.

The k factor depends on the specimen geometry. As an example, according to Greensmith (1963)in the case of Single Edge Notched Tension (SENT) specimens, this parameter is expressed as follows:

kðkÞ ¼2:95  0:08ð1  kÞffiffiffi k

p ð3Þ

This result, based upon experimental data, was con-firmed by a finite element (FE) analysis by Lindley

(1972). Let us note that the energy interpretation of the J

integral introduced by Rice (1968) and Cherepanov

(1968) is equivalent to the strain energy release rate G

for elastic materials. Thus, the fracture energy can be also evaluated by using the J integral. The fracture mechanics properties are dependent on the sharpness of the notch

(Trapper and Volokh, 2008; Volokh, 2013) and this

influ-ence decreases when large strain is involved since the crack radius increases to form a smooth notch (

Naït-Abdelaziz et al., 2012).

Dealing with multiaxial fracture of rubbers, in the case of circular defects,Naït-Abdelaziz et al. (2012)have pro-posed a general form for T which accounts for the biaxiality ratio at large strain:

T ¼ G ¼ J ¼ 2kceqgðkeqÞWa ð4Þ

in which

c

is a parameter depending on the biaxiality ratio ranging approximately from 1 (uniaxial tension) to 1.4 (equibiaxial tension) and keqis the equivalent stretch ratio

expressed as a function of the principal stretches:

keq¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k21þ k 2 2þ k 2 3 3 s ð5Þ in which k1, k2and k3are the principal stretches.

The function g in Eq. (4) depends on the equivalent stretch ratio and can be fitted by the following equation: gðkeqÞ ¼ 0:255 þ2:837 ðkeqÞ2 2:888 ðkeqÞ4 þ2:507 ðkeqÞ6 ð6Þ

2.2. Constitutive mechanical models

The basic features of the rubber stress–strain response are generally described by large strain elastic constitutive models which can be classified into two categories: the first one, physically-based, is issued from statistical mechanics theories (Treloar, 1943; James and Guth,

1943; Arruda and Boyce, 1993) and the second one,

phe-nomenological-based, is issued from invariant-based and stretch-based continuum mechanics approaches (Rivlin,

1948; Ogden, 1984; Yeoh, 1990). The invariant-based

Rivlin (1948)and stretch-basedOgden (1984)

phenomeno-logical models are probably the most popular and are expressed in terms of strain energy density (SED) func-tions, respectively, as follows:

1

It is based upon the assumption that defects always exist in a given material and act as stress concentrators. Consequently, the material failure is the consequence of the growth of virtual cracks.

W ¼X 1 i¼0 X1 j¼0 CijðI1 3Þ i ðI2 3Þ j ð7Þ W ¼X 1 i¼1

l

i

a

i ðkai 1 þ k ai 2 þ k ai 3  3Þ ð8Þ

where I1and I2are the two first invariants of the right

Cau-chy–Green strain tensor. The terms Cij,

a

iand

l

iare

mate-rial parameters to be identified experimentally.

Let us now highlight the particular case of a first-order development in the invariant-based Rivlin (1948)model (i.e. j = 0 and i = 1). In this case, the SED function given in Eq.(7)reduces to:

W ¼ C10ðI1 3Þ ¼ C10ðk21þ k 2 2þ k

2

3 3Þ ð9Þ

Note that a first-order development of the stretch-basedOgden (1984)model given in Eq.(8)leads to: W ¼

l

1

a

1ðk a1 1 þ k a1 2 þ k a1 3  3Þ ð10Þ

which reveals that Eqs.(9) and (10)are formally equivalent if

a

1= 2 and C10=

l

1/2. For this particular case, the

param-eter

l

1is the so-called shear modulus.

A physically-based approach (Treloar, 1943), describing the kinematics of a single macromolecular chain in the framework of the statistical mechanics allowed to get, for a perfect network, the so-called Neo-hookean model. In this statistical Gaussian model which represents the fun-dament of the rubber elasticity, the SED function is expressed as follows: W ¼nkT 2 ðk 2 1þ k 2 2þ k 2 3 3Þ ð11Þ

where n is the density of the elastically active chains, k is the Boltzmann constant and T is the absolute temperature which underlines the entropic nature of the rubber elastic-ity. Comparing Eqs.(9)–(11)it comes that:

l

1¼ 2C10¼ nkT ð12Þ

The previous model assumes a perfect network (also called affine network). Particularly, in this approach, the terminations of the network elastic segments are supposed to be fixed. In a real network, these terminations play the role of junction nodes of different elastic segments (cross-linking nodes for example) and can therefore move around a mean position. To account for the mobility of these junc-tion nodes,James and Guth (1943)modified the Neo-hook-ean model and derived the so-called phantom network model expressed as follows:

W ¼nkT 2 1  2 f   ðk2 1þ k 2 2þ k 2 3 3Þ ð13Þ

In Eq.(13), f is the crosslink functionality (i.e. the num-ber of the elastically active chains linked to the same cross-link node) whose a common value is 3 or 4 for rubbers.

It can be alternatively written as follows: W ¼

q

RT 2Mc 1 2 f   ðk21þ k 2 2þ k 2 3 3Þ ð14Þ

and, when considering uniaxial tension: W ¼

q

RT 2Mc 1 2 f   k2þ2 k 3   ð15Þ where R is the perfect gas constant,

q

is the reference mass density for a given state of ageing and Mcis the average

molar mass of the elastically active chains (i.e. between crosslinks).

2.3. Intrinsic defect concept

Perfect materials do not exist in nature, and all materi-als generally contain flaws issued from processing or pre-ageing (such as voids and micro-cracks). Rubbers are gen-erally reinforced by particles and mainly by carbon black. This filler is generally very small (its size is lower than 100 nm). Nevertheless, it can form aggregates of which the size can reach a few microns. These aggregates can play a role of intrinsic defects which act as potential cracks. They directly affect the strength (Bueche, 1959; Roland

and Smith, 1985) or the fatigue life properties (Braden

and Gent, 1960; Gent et al., 1964; Choi and Roland, 1996).

The approach we have developed in Naït-Abdelaziz

et al. (2012)is founded on the assumption that the failure

of a smooth specimen loaded in tension could be attributed to the propagation of the intrinsic defect. So, for a given defect size, if the fracture toughness in terms of Jcis

avail-able, then we can derive the stretch ratio at failure from Eq.

(4). In this study, the intrinsic defect was calculated by using Eq.(4), the inputs being the fracture energy Jc, the

stretch ratio at break of a smooth specimen in tension kc

and the SED function W. From this approach, we have esti-mated the stretch ratios at break for different loading paths (i.e. biaxiality ratios) and found a very nice agreement with experimental data for a Natural Rubber and a Styrene Buta-diene Rubber.

The objective of this contribution is to estimate the stretch ratios at break under tension loading for various thermal ageing conditions. The adopted approach is to cal-culate the intrinsic defect for a virgin (unaged) material and to derive the stretch ratios for the thermally aged materials, the Jc value and also the SED function being

available for each time of exposure. This methodology will be detailed further.

2.4. Time–temperature equivalence

The time–temperature principle was specifically inves-tigated to address the issue of the similarity in the evolu-tion of the mechanical behaviour of viscoelastic materials (such as polymers) when increasing the strain rate or decreasing the temperature (Treloar, 1971). It consists in constructing a so-called master curve from the measure-ments of a mechanical property (modulus for example) in a given range of temperatures (or strain rates) for differ-ent strain rates (or temperatures). The obtained curves can be then superimposed by shifting them via a shift factor noted aT to get the master curve for the mechanical

property under investigation. Therefore, this master curve allows to estimate the mechanical property for a

temperature or a strain rate value (not experimentally measurable) by extrapolation. The shift factor can be fitted using an Arrhenius type law which takes the following form: lnðaTÞ ¼  Ea R 1 T 1 T0   ð16Þ where Eais the activation energy, T is the absolute

temper-ature, T0is the reference temperature and R is the perfect

gas constant.

This form is generally used when T0> Tg+ 100, Tgbeing

the glass transition temperature of the polymer. This model has been widely and successfully used to investigate the thermal ageing of polymers (Ha-Anh and Vu-Khanh,

2005; Gillen et al., 2006; Woo and Park, 2011). When the

previous restriction on the reference temperature is no longer valid, one may use the phenomenological approach developed by Williams–Landel–Ferry (Williams et al., 1955), known as the WLF approach. According to these authors, the shift factor can be approximated by the fol-lowing expression:

lnðaTÞ ¼

C1ðT  T0Þ

C2þ T  T0 ð17Þ

in which C1and C2are two adjustable parameters

depend-ing on the material under study.

3. Material and experimental procedure 3.1. Material

The material investigated in this work is an Ethylene– Propylene–Diene Monomer (EPDM) rubber. It consists in the following molar fractions of monomer units: 66.1 mol.% ethylene, 33.1 mol.% propylene and 0.8 mol.% norbornene (ENB). It is vulcanised by 4 phr of sulphur com-pounds and filled by 13 and 29.8 wt% of carbon black and clay platelets, respectively.

3.2. Specimens

Three specimen kinds were used in this work. They were cut from EPDM sheets. The average molar mass mea-surements were achieved using circular flat discs. Dogbone specimens were selected to achieve tensile tests and get the stress–strain relationship and the ultimate properties. Double Edge Notched Tension (DENT) specimens were chosen to measure the fracture energy. Note that the thick-ness of these three specimen kinds is the same and equal to 3.8 mm.Fig. 1shows the specimen geometries.

3.3. Accelerated ageing procedure

The three specimen kinds were exposed in air-venti-lated ovens at 130, 150 and 170 °C for various exposure times. Let us note that a dedicated set-up was designed in order to keep the dogbone and DENT specimens in a ver-tical position and thus, avoid any change in form during their exposure to high temperature in the air-ventilated ovens.

3.4. Average molar mass measurements

For each test condition in terms of ageing temperature and exposure time, the average molar mass of the elasti-cally active chains Mcwas determined by swelling in

cyclo-hexane at 25 °C in a Soxhlet apparatus. Indeed, when a rubbery polymer network is placed in a suitable solvent (presenting a good chemical affinity with the polymer), the polymer tends to absorb the maximum of solvent in its free volume (Fig. 2a). This reversible physical process causes the network volume expansion in the three space directions as shown inFig. 2b.

This swelling ability depends on the interactions between the polymer chains and solvent molecules, in addition to the length of polymer chains between cross-links which is defined by the average molar mass of the elastically active chains Mc. This parameter is given by

the Flory–Rehner relationship for a 4-functional network

(Marzocca, 2007): Mc¼  0:5V

q

polymerðV 1=3 r0  0:5Vr0Þ lnð1  Vr0Þ þ Vr0þ

v

V2r0 ð18Þ

where V is the molar volume of the solvent (cm3_/mol),

q

polymer is the polymer density (0.86 g/cm3 for EPDM),

v

is the Flory–Huggins interaction parameter between the polymer and the solvent equal to 0.321 for EPDM-cyclo-hexane (Baldwin and Ver Strate, 1972; Hilborn and

Ranaby, 1989) and Vr0is the polymer volume fraction in

the swollen network expressed as:

Vr0¼

q

solvent

q

solventþ ðmg=ms 1Þ

q

polymer

ð19Þ

where mgis the weight of swollen polymer sample, msis

the weight of the same sample after drying under vacuum at 40 °C for 24 h and

q

solventis the solvent density (0.78 g/

cm3_{for cyclohexane).}

3.5. Tensile and fracture mechanics tests

Monotonic tensile tests were performed in order to get the stress–strain relationship and the ultimate properties (in terms of strain and stress at break). Tests were achieved until fracture at room temperature (25 °C) and under a constant crosshead speed of 5 mm/mn on an electrome-chanical Instron 5800 set-up equipped with a load cell of 1 kN. Strains were measured using a non-contact video-system allowing to track the displacement of two spots previously printed on the dogbone specimens. For each exposure time and temperature, seven tests were carried out to estimate the natural scattering of the mechanical response.

Fracture tests were achieved in order to estimate the fracture toughness in terms of fracture energy Gcby using

the DENT specimens. They were performed with the same apparatus used for the tensile tests at room temperature and under a constant crosshead speed of 15 mm/min.

4. Experimental results 4.1. Average molar mass

The average molar mass dependence on both tempera-ture and exposure time is shown inFig. 3a. As expected, increasing the exposure time leads to the decrease of the average molar mass while increasing the temperature accelerates this process. Using Eq.(16), the time–tempera-ture equivalence was applied to the average molar mass data. The best shift factor, for a reference temperature of 403 K, was obtained by using an activation energy equal to 104 kJ/mol. This value is in the same order of magnitude than that given by Ajalesh Balachandran et al. (2012). It appears that the average molar mass could be taken as a parameter reflecting the degradation state of the material. The evolution of the average molar mass as a function of the reduced time taT, shown inFig. 3b, can be fitted by a

decreasing exponential law of the form:

Mc¼ b þ degtaT ð20Þ

where b, d and

g

are constants determined using a least square method which values are b= 600 g/mol, d= 1100 g/mol and

g

= 9.5  104_h1_.

4.2. Ultimate properties

From the uniaxial tensile tests, failure strain and stress were measured for each ageing condition (temperature and exposure time). As expected, either the failure strain or the failure stress decreases when increasing the expo-sure time at a given temperature. This decrease is dramat-ically amplified when increasing the temperature.Fig. 4

shows the changes in these ultimate properties as a func-tion of the reduced time, after applying the time–temper-ature equivalence principle. Note that the same value of the activation energy as that calculated for the average molar mass was used to shift these data. That is to say that the same value of the shift factor was used in this case. In

Fig. 4a is shown the evolution of the failure stretch ratio

while inFig. 4b the true stress at break is reported. As clearly shown in these figures, the time–temperature equivalence allows to aggregate the experimental data and to build a master curve for each ultimate property. 4.3. Fracture energy

The fracture energy Jcwas computed from the

experi-mental data using Eqs.(2) and (3). A FE analysis on a DENT specimen using the MSC.Marc software also allowed to compute the critical J integral. A comparison is shown in

(a)

(b)

(c)

3.8 mm 25.4 mm a W = 25 mm H = 60 mm a

Fig. 1. Specimen geometries for: (a) average molar mass measurements, (b) tensile and (c) fracture mechanics tests.

(a) (b)

Fig. 5where the FE results are plotted against the experi-mental data. All the dots lay around the bisectrix indicating a quite nice agreement.

The whole data are then reported inFig. 6as a function of the average molar mass square root and can be fitted by a linear law of the form:

Jc¼ A ffiffiffiffiffiffiffiffi Mc0 p ffiffiffiffiffiffiffiMc p  ffiffiffiffiffiffiffiffiMc0 q   ð21Þ where A and Mc0 are constants which values, given by a least square method, are 125 kJ/m2_{and 680 g/mol,}

respec-tively. This linear dependence on the average molar mass square root was reported for rubbers, but exclusively for the threshold fracture energy (Lake and Thomas, 1967). 4.4. Stress–strain relationship

Tensile tests allowed to obtain the evolution of the true stress as a function of the true strain.Fig. 7shows an exam-ple of these relationships for an ageing temperature of 130 °C and for different exposure times. This figure clearly highlights that both the modulus and the hardening increase with respect to the ageing duration. Although only results for one temperature are given, the same trends were observed in the other studied temperatures.

The first-order Ogden SED function, given in Eq.(10), was used to describe the changes in the stress–strain

relationship. This choice is reasonable and leads to a quite good description of our experimental results, as shown in

Fig. 8for particular examples.

The parameters

l

and

a

of the Ogden model (Note that the suffix 1 is suppressed) were determined for each age-ing condition.Fig. 9shows the evolution of the shear mod-ulus

l

as a function of the average molar mass. Are also plotted the evolution of the theoretical modulus in the case of the phantom network for two crosslink functionality values. It is worth noting that the experimental data are bounded by the theoretical phantom network model.

The exponent

a

is then plotted as a function of the aver-age molar mass inFig. 10. This parameter decreases when increasing the average molar mass. Its value ranges

0 400 800 1200 1600 2000 0 500 1000 1500

t (h)

M

c

(g/mol)

130°C 150°C 170°C 0 400 800 1200 1600 2000 0 1000 2000 3000 4000

ta

T

(h)

M

c

(g/mol)

130°C 150°C 170°C Eq. (20)

(a)

(b)

Fig. 3. Average molar mass as a function of: (a) the exposure time and (b) the reduced time, for three temperatures under study.

0 3 6 9 12 0 1000 2000 3000 4000 taT(h) λ 130°C 150°C 170°C

(a)

(b)

0 20 40 60 80 0 1000 2000 3000 4000 c (MP a) ta_T (h) 130°C 150°C 170°C

σ

Fig. 4. Ultimate properties as a function of the reduced time: (a) failure stretch ratio and (b) failure stress.

0 20 40 60 80 100 0 20 40 60 80 100 Jc (exp) (kJ/m 2 ) Jc (FE) (kJ/m 2 ) 130°C 150°C 170°C bisectrix

between 3 and 2, this latter value corresponding to a Neo-hookean model.

5. Prediction of the ultimate properties

In this section, we plan to build a prediction criterion by combining the intrinsic defect concept with the fracture mechanics approach. The main idea is that a material is

never perfect and contains defects or heterogeneities (aggregates of reinforcement particles usually carbon black, voids, or defects induced during processing or pre-ageing, etc.). These pre-existing defects can act as stress concentrators when loading a specimen. The intrinsic defect approach is therefore used to predict the failure of smooth specimens. This approach was successfully used to predict the biaxial fracture of a Natural Rubber and a Styrene Butadiene Rubber (Naït-Abdelaziz et al., 2012). The prediction of the failure stretch ratios is achieved by using Eq.(4). In this equation, Jcis given by Eq.(21), ath

is the theoretical defect size and W is the SED. Knowing the critical value of J, and assuming that the failure of a specimen without crack is due to an intrinsic defect, it is possible to calculate the size of this defect which is assumed to be an intrinsic quantity of the material and thus equal to a constant. Moreover, the degradation of the macroscopic material behaviour is a direct conse-quence of the changes in the material microstructure. In this work, we assume that the dominant parameter that governs the mechanical properties, and is a pertinent indi-cator of the macromolecular network degradation, is the average molar mass of the elastically active chains Mc. So,

capturing the evolution of this property with the exposure time would allow to predict the evolution of the macro-scopic mechanical properties.

0 20 40 60 80 20 30 40 50 60 70 Mc 1/2 (g/mol)1/2 Jc (kJ/m²) 130°C-exp 130°C-FE 150°C-exp 150°C-FE 170°C-exp 170°C-FE Eq. (21)

Fig. 6. Fracture energy as a function of the average molar mass square root. 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 true strain

true stress (MPa)

0 h 504 h 1112 h 1614 h 2184 h 3024 h

Fig. 7. Changes in the stress–strain relationship for an ageing tempera-ture of 130 °C. 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 true strain

true stress (MPa)

0 h 176 h

576 h

Fig. 8. Comparison between first-order Ogden model (dashed lines) and experiments (solid lines).

1 2 3 4 0 500 1000 1500 2000 2500 Mc (g/mol) α 130°C 150°C 170°C

Fig. 10. Exponentaas a function of the average molar mass.

0 1 2 3 4 5 6 0 500 1000 1500 2000 2500

(MP

a)

M

c

(g/mol)

130°C 150°C 170°C phantom network: f=4 phantom network: f=3

μ

5.1. Determination of the intrinsic defect size

Considering the virgin material, the fracture energy at room temperature is first determined using a DENT speci-men. For the DENT specimen (Fig. 1c), the fracture energy (equivalent to the J integral) could be written as:

J ¼ T ¼ 4kW0a ð22Þ

Note that it is twice the value of the SENT specimen given in Eq.(2)because the specimen contains two cracks. In Eq.(22), W0is the SED of the virgin material, a is the

crack length (5 mm in our case) and k is a decreasing func-tion of the applied stretch ratio k according to (Lake and

Thomas, 1967):

k ¼

p

ffiffiffi k

p ð23Þ

By using Eq.(22), we have obtained a value of 66 kJ/m2 for Jc. A FE analysis on the same specimen gives a critical J

integral value of 70 kJ/m2_{which is in good agreement with}

the experimental value. Thus, the failure stretch ratio of a virgin smooth specimen being available, the theoretical defect size athcan be computed using Eq.(4). The value

we have obtained is 230

l

m for Jc= 66 kJ/m2. If we use

the FE result of Jc (70 kJ/m2), the value of ath is 250

l

m

(radius of a circular defect in a tensile smooth specimen). 5.2. Prediction of the failure stretch ratio

As mentioned above, the degradation of the material properties is assumed to be strictly linked to the evolution of the average molar mass. The prediction of the failure stretch ratios is achieved by using Eq.(4). In this equation, Jcis given by Eq.(21), athis the virtual defect size already

estimated and W is the SED. This SED can be calculated by using the Ogden formula given by Eq. (10)since the model parameters are directly related to the average molar mass Mc. To simplify the procedure, we assume, in what

follows, a phantom network model which implies that the shear modulus is given by Eq. (13) with a crosslink functionality equal to 4. The exponent

a

is considered as a constant and equal to 2.

When the fracture occurs, Eq. (4) can therefore be rewritten as follows: A ffiffiffiffiffiffiffiffi Mc0 p ffiffiffiffiffiffiffiMc p  ffiffiffiffiffiffiffiffiMc0 q   ¼ 2kceqcgðkeqcÞ 

q

RT 2Mc 1 2 f   k2cþ 2 kc 3   ath ð24Þ where keqc¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðk2cþ 2k1c Þ=3 q

in uniaxial tension and g is given by Eq.(6). The exponent

c

is taken equal to 1. The suffix c denotes the critical value, i.e. that corresponding to the specimen fracture.

Since Eq.(24)is non-linear, it is solved numerically to estimate the failure stretch ratio kc for a given value of

the average molar mass. Moreover, the average molar mass being linked to the reduced time by Eq.(20), it is therefore possible to plot the estimated failure stretch ratio as a function of the reduced time. The predicted evolution is compared to the experimental data inFig. 11. A very nice agreement is obtained confirming that the intrinsic defect concept combined with the fracture mechanics can be used to derive a failure criterion accounting for the degradation due to ageing.

5.3. Prediction of the failure stress and the critical strain energy density

The Cauchy stress tensor components can be derived from the SED function W. In the principal directions, it is

0 3 6 9 12 0 1000 2000 3000 4000 taT(h) λ 130°C 150°C 170°C criterion

Fig. 11. Prediction of the failure stretch ratio as a function of the reduced time. 0 20 40 60 80 0 1000 2000 3000 4000 taT(h) σc (MPa) 130°C 150°C 170°C criterion 0 10 20 30 40 0 500 1000 1500 2000 2500 3000 3500 4000 taT(h) W c (MPa) 130°C 150°C 170°C criterion

(a)

(b)

Fig. 12. Prediction of the failure stress (a) and the critical SED (b) as a function of the reduced time.

given, for an incompressible material, by the following formula:

r

i¼ ki

@W

@ki þ p ð25Þ

where p is the hydrostatic pressure determined by using the boundary conditions of the problem. For example, under uniaxial tension, only one component takes a non-zero value. In this case, the term p is determined using for example the condition

r

2= 0. Using the SED function

given by Eq.(15), the Cauchy stress under uniaxial tension is expressed as follows:

r

¼

q

RT Mc 1 2 f   k21 k   ð26Þ The failure stress is obtained when the stretch ratio takes its critical value kcpreviously estimated. Using the

same time–temperature equivalence derived from the average molar mass evolution, the failure stress can be plotted as a function of the reduced time as shown in

Fig. 12a. The experimental data are in good agreement

with the estimated stresses derived from the intrinsic defect criterion. Finally, as shown inFig. 12b, we can also estimate the critical strain energy density by using Eq.

(15)provided that k is replaced by kc.

6. Conclusion

The objective of this study was to establish a failure cri-terion accounting for degradation due to ageing in the case of rubbers. First, from accelerated ageing tests, we have experimentally determined the changes in the ultimate mechanical properties such as the failure stretch ratios and stresses or the fracture energy in terms of critical J integral. Since degradation due to the thermal oxidation results in a competitive process of post-crosslinking versus chain scission, it is necessary to introduce microstructure features in the description of the material behaviour. The average molar mass of the elastically active chains (i.e. between crosslinks) was found to be a good indicator of degradation of the macromolecular network. Moreover, using the time–temperature equivalence combined with the Arrhenius model, we have shown that a master curve representing the evolution of the average molar mass as a function of a reduced time could be obtained. The activa-tion energy was found in the same order of magnitude than that generally given in the literature. Using this acti-vation energy and applying the same model, master curves were derived for the ultimate mechanical properties such as failure stretch ratios and stresses or critical strain energy density.

On the other hand, fracture mechanics tests on DENT specimens allowed to determine the fracture energy in terms of critical J integral which was found linearly depen-dent on the average molar mass square root. Finally, tensile tests allowed to derive the stress–strain relationship which can be fitted by using a first-order Ogden model. We have shown that, in a first approximation, the phantom network could be alternatively used to describe the material behav-iour, provided that the functionality junction parameter is

set to 4. Such a description allows to introduce the average molar mass as the main influent parameter on the mechanical response.

Combining the intrinsic defect concept with the frac-ture mechanics, we can therefore derive prediction of the ultimate properties. This approach requires as inputs, the average molar mass master curve, the evolution of the crit-ical J integral versus the average molar mass and the intrin-sic defect size. It also requires the stress–strain relationship which could be, as a first approximation, given by the phantom network theory. Finally, we have shown that the predicted values for all the ultimate properties are in good agreement with the experimental data.

Using such a method allows to predict time to failure for lower operating temperatures and makes therefore this tool very attractive when designing rubber components subjected to thermal oxidative environment.

Even the failure analysis was only based upon uniaxial tension and monotonic loading, it could be extended to any biaxial monotonic loading by using the approach we have developed in a previous paper (Naït-Abdelaziz et al., 2012). For non-monotonic loading paths, it will be neces-sary to account for the viscous effects. Such a modeling is actually in progress.

References

Ajalesh Balachandran, N., Kurian, P., Rani, J., 2012. Effect of aluminium hydroxide, chlorinated polyethylene, decabromo biphenyl oxide and expanded graphite on thermal, mechanical and sorption properties of oil-extended ethylene–propylene–diene terpolymer rubber. Mater. Des. 40, 80–89.

Arruda, E.M., Boyce, M.C., 1993. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41, 389–412.

Assink, R.A., Gillen, K.T., Sanderson, B., 2002. Monitoring the degradation of a thermally aged EPDM terpolymer by 1_{H NMR relaxation}

measurements of solvent swelled samples. Polymer 43, 1349–1355.

Baldwin, F.P., Ver Strate, G., 1972. Polyolefin elastomers based on ethylene and propylene. Rubber Chem. Technol. 45, 709–881.

Braden, M., Gent, A.N., 1960. The attack of ozone on stretched rubber vulcanizates. II. Conditions for cut growth. J. Appl. Polym. Sci. 3, 100– 106.

Bueche, F., 1959. The tensile strength of elastomers according to current theories. Rubber Chem. Technol. 32, 1269–1285.

Cherepanov, G.P., 1968. Cracks in solids. Int. J. Solids Struct. 4, 811–831.

Choi, I.S., Roland, C.M., 1996. Intrinsic defects and the failure properties of cis-1,4 polyisoprenes. Rubber Chem. Technol. 69, 591–599.

Clavreul, R., 1997. Evolution of ethylene propylene copolymers properties during ageing. Nucl. Instrum. Methods Phys. Res., Sect. B 131, 192– 197.

Colin, X., Audouin, L., Verdu, J., 2007a. Kinetic modelling of the thermal oxidation of polyisoprene elastomers. Part III – oxidation induced changes of elastic properties. Polym. Degrad. Stab. 92, 906–914.

Colin, X., Audouin, L., Verdu, J., 2007b. Kinetic modelling of the thermal oxidation of polyisoprene elastomers. Part I – unvulcanized unstabilized polyisoprene. Polym. Degrad. Stab. 92, 886–897.

Colin, X., Audouin, L., Le Huy, M., Verdu, J., 2007c. Kinetic modelling of the thermal oxidation of polyisoprene elastomers. Part II – effect of sulfur vulcanization on mass changes and thickness distribution of oxidation products during thermal oxidation. Polym. Degrad. Stab. 92, 898–905.

Ferry, J.D., 1970. Viscoelastic properties of polymers. Br. Polym. J., New York: John Wiley & Sons, Ltd.

Gent, A.N., Lindley, P.B., Thomas, A.G., 1964. Cut growth and fatigue of rubbers. Part I: the relationship between cut growth and fatigue. J. Appl. Polym. Sci. 8, 455–466.

Gillen, K.T., Bernstein, R., Clough, R.L., Celina, M., 2006. Lifetime predictions for semi-crystalline cable insulation materials: I. Mechanical properties and oxygen consumption measurements on EPR materials. Polym. Degrad. Stab. 91, 2146–2156.

Greensmith, H.W., 1963. Rupture of rubbers. X. The change in stored energy on making a small cut in a test piece held in simple extension. J. Appl. Polym. Sci. 7, 993–1002.

Griffith, A.A., 1921. The phenomenon of rupture and flow in solids. Philos. Trans. R. Soc. J. Ser. A 221, 163–198.

Ha-Anh, T., Vu-Khanh, T., 2005. Prediction of mechanical properties of polychloroprene during thermo-oxidative aging. Polym. Testing 24, 775–780.

Hilborn, J., Ranaby, B., 1989. Photocrosslinking of EPDM elastomers. Photocrosslinkable compositions. Rubber Chem. Technol. 62, 592– 608.

James, H.M., Guth, E., 1943. Theory of elastic properties of rubbers. J. Chem. Phys. 11, 455–481.

Lake, G.J., Thomas, A.G., 1967. The strength of highly elastic materials. Proc. R. Soc. Ser. A 300, 108–119.

Lindley, P.B., 1972. Energy for crack growth in model rubber components. J. Strain Anal. Eng. Des. 7, 132–140.

Marzocca, A.J., 2007. Evaluation of the polymer-solvent interaction parametervfor the system cured styrene butadiene rubber and toluene. Eur. Polym. J. 43, 2682–2689.

Naït-Abdelaziz, M., Zaïri, F., Qu, Z., Hamdi, A., Aït Hocine, N., 2012. J integral as a fracture criterion of rubber-like materials using the intrinsic defect concept. Mech. Mater. 53, 80–90.

Ogden, R.W., 1984. Non linear elastic deformation. Ellis-Horwood Limited Publishers, England.

Rajeev, R.S., De, S.K., Bhowmick, A.K., John, B., 2003. Studies on thermal degradation of short melamine fibre reinforced EPDM, maleated EPDM and nitrile rubber composites. Polym. Degrad. Stab. 79, 449– 463.

Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386.

Rivaton, A., Cambon, S., Gardette, J.L., 2005. Radiochemical ageing of EPDM elastomers. 3. Mechanism of radiooxidation. Nucl. Instrum. Methods Phys. Res., Sect. B 227, 357–368.

Rivlin, R.S., 1948. Large elastic deformations of isotropic materials. Philos. Trans. R. Soc. Ser. A 241, 459–525.

Rivlin, R.S., Thomas, A.G., 1953. Rupture of rubber. Part 1: characteristic energy for tearing. J. Polym. Sci. 10, 291–318.

Roland, C.M., Smith, C.R., 1985. Defect accumulation in rubber. Rubber Chem. Technol. 58, 806–814.

Tomer, N.S., Delor-Jestin, F., Singh, R.P., Lacoste, J., 2007. Cross-linking assessment after accelerated ageing of ethylene propylene diene monomer rubber. Polym. Degrad. Stab. 92, 457–463.

Trapper, P., Volokh, K.Y., 2008. Cracks in rubber. Int. J. Solids Struct. 45, 6034–6044.

Treloar, L.R.G., 1943. The elasticity of a network of long chain molecules. Trans. Faraday Soc. 39, 36–41.

Treloar, L.R.G., 1971. Viscoelastic properties of polymers. Br. Polym. J., London: John Wiley & Sons, Ltd.

Volokh, K.Y., 2013. Review of the energy limiters approach to modeling failure of rubber. Rubber Chem. Technol. 86, 470–487.

Williams, M.L., Landel, R.F., Ferry, J.D., 1955. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J. Am. Chem. Soc. 77, 3701–3707.

Woo, C.S., Park, H.S., 2011. Useful lifetime prediction of rubber component. Eng. Fail. Anal. 18, 1645–1651.

Yeoh, O.H., 1990. Characterization of elastic properties of carbon black filled rubber vulcanizates. Rubber Chem. Technol. 63, 792–805.